that's a different game from putting your life at risk.
I agree. It is however a strong motivation that has an effect on the choices of the civilian leadership of the military.
That is straight up at odds with American (or Canadian, or for the most part, UK) history. In the United States Trump is the first "war-time" president to blow re-election during an active war, and his term was punctuated by attempts at troop draw down, coupled with inflammatory rhetoric towards Iran when he was low in polls.
Engaging in something that can be spun as a just war is a viable re-election tactic.
The important part here is that they are forced to actually take the steps of justifying the war to the public and consulting the legislature.
If you have the public’s support for the war, then a war isn’t going to hurt your election chances.
I'm not sure I'd count Trump as a wartime President, though perhaps that's a result of appallingly low standards for what counts as peace.
It doesn’t stop them from going to war, but it makes it less likely.